These two functions evaluate the incomplete elliptic integral of the third
kind Π(n, φ, k) and its complete counterpart Π(n,
k) = E(n, π/2, k).

The return type of these functions is computed using the result
type calculation rules when the arguments are of different
types: when they are the same type then the result is the same type as
the arguments.

These functions are computed using only basic arithmetic operations, so
there isn't much variation in accuracy over differing platforms. Note that
only results for the widest floating point type on the system are given
as narrower types have effectively zero error.
All values are relative errors in units of epsilon.

The functions are then implemented in terms of Carlson's integrals using
the relations:

and

[1]
I haven't been able to find a literature reference for this relation,
but it appears to be the convention used by Mathematica. Intuitively
the first 2 * m * Π(n, k) terms cancel out as the
derivative alternates between +∞ and -∞.